I ask this in relation to ” /(x/y) ” = ” *(y/x) ”
My mathematical ignorance does not allow me to perceive exactly what it is that confuses me about these manoeuvres and so perhaps my question is vague.
I have no difficulty with it as a technique; as something through which I can put an expression, and out at the other end the right result will appear. What I am trying to understand is *why it works*, contrasted with remembering it as a kind of magical spell.
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**EDIT:**
It was very rewarding for me to read all of your comments. Thank you most kindly for enlightening me.
For those interested in the cause of my previous confusion:
The gaps in my understanding of going from y*x=z to y=z/x were definitions of the equal sign and division.
I can see now that I previously considered the = sign to mean «result» or «answer» in some sort of final sense, like a conclusion; I now see that it only states that this is equal to that.
Following this fundamental piece of knowledge, I can belatedly understand what an equation is. From there, via the definition of division as the opposite of multiplication, I can see that if I divide something while also multiplying it with the same number, these actions cancel each other out.
And so the magical spell between y*x=z and y=z/x is the logic above expressed mathematically as x/(y*x)=z/x.
In: 25
Since the left side is equal to the right side, if you perform the same operation to both sides, then both sides will remain equal. They’ll still be equal even if you replace any term with an equivalent term or expression.
An equation is a sentence that states a fact. Like, *Apples are good for you.*
That’s a sentence that states a fact.
What’s different about equations from other sentences is they always refer to quantities (numbers).
And all equations work the same way. They say that what’s on one side of the equal sign is exactly the same quantity as what’s on the other side.
This means that if you change one side, you have to change the other in the exact same way. Add a million. Divide by 12. Multiply by 87. Whatever.
Imagine the equation as a set of scales. Everything to the left of the = sign is on one side, everything on the right of the = sign is on the other. Now you don’t know the actual weight on each side because you’re using variables instead of numbers, but you do know that both sides weigh the same.
What happens if we take away half the weight on the left side? The right side now weighs twice as much since they were the same weight in the beginning. To equal out these scales, you’ll have to take away half of the right side’s weight. This applies to any change in weight. If you add, subtract, multiply, or divide one side’s weight, you need to do the same thing to the other side to keep it balanced.
Applying this to your example, y*x somehow became just y. This happened by dividing y*x by x, cancelling the two x’s out. Well, to make that change valid, you have to also divide the right side by x. This leads you to y = z/x.
This “rebalancing the scales” is the central premise behind algebra.
Symbolically:
y * x = z <- Equation 1
If you do the same thing to both sides of an equation, the equality still holds. Divide both sides of (Equation 1) by x:
(y * x )/ x = (z )/ x <- Equation 2
But (y*x)/x = y * x / x . Now, x / x = 1. and y * 1 = y, therefore Equation 2 simplifies to:
y = z/x
We can think of multiplication as addition on steroids. Similarly, we can think of division as subtraction on steroids.
Any multiplication can be expressed as a series of additions. For example, 3*5 is a concise way of expressing 3+3+3+3+3. “Three times five” means “the sum of five instances of the value 3”.
To reverse the effect of addition, we can subtract. Start with 3; add 5 to get 8. Start with 8. Subtract 5 to get 3. (3+5=8, 8-5=3). We don’t often do this, but we can even think of the subtraction expression in terms of addition: what value, when added to five, produces the sum of eight.
In a smilar fashion we can reverse the effect of multiplication by dividing. Start with 3; multiply by 5 to get 15. Start with 15; divide by 5 to get 3. We can express the division in terms of multiplication: what value, multiplied by 3, produces the product of 15.
In mathematics division is defined using multiplicative inverse. It is possible to inspect equations of the form a*x = b without defining multiplicative inverse. In this case you have definition for your own special multiplication *. You can solve equations involving your special multiplication by trial and error. When you introduce the concept of an inverse, solving these kind of equations becomes straightforward. There is no obvious connection why inverses makes solving equations straightforward meaning that there might be other ways to solve equations using some algorithm. We are fortunate that we can visualize numbers using number line and simultaneously solve equations of the form a*x = b using inverse elements which we can see in the number line. That is a fortunate coincidence.
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